In the present paper we establish the Wiener test for boundary regularity of the solutions to the polyharmonic operator. We introduce a new notion of polyharmonic capacity and demonstrate necessary and sufficient conditions on the capacity of the domain responsible for the regularity of a polyharmonic function near a boundary point. In the case of the Laplacian the test for regularity of a boundary point is the celebrated Wiener criterion of 1924. It was extended to the biharmonic case in dimension three by Mayboroda and Maz’ya (Invent Math 175(2):287–334, 2009). As a preliminary stage of this work, in Mayboroda and Maz’ya (Invent Math 196(1):168, 2014) we demonstrated boundedness of the appropriate derivatives of solutions to the polyharmonic problem in arbitrary domains, accompanied by sharp estimates on the Green function. The present work pioneers a new version of capacity and establishes the Wiener test in the full generality of the polyharmonic equation of arbitrary order.
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Acknowledgements The first author was in part supported by the Alfred P. Sloan Fellowship, NSF grant DMS 1344235, NSF grant DMS 1220089, UMN MRSEC Seed grant DMR 0212302, and Simons Fellowship in Mathematical Sciences. The second author was partly supported by the Ministry of Education and Science of the Russian Federation, agreement n. 02.a03.21.0008.